High pressure phonon and thermodynamic properties of Ru based intermetallics: A DFT investigation

Accepted Manuscript High pressure phonon and thermodynamic properties of Ru based intermetallics: A DFT investigation Ekta Jain, Gitanjali Pagare, Shubha Dubey, Ramesh Sharma, Yamini Sharma PII:

S0022-3697(18)30357-3

DOI:

10.1016/j.jpcs.2018.06.036

Reference:

PCS 8650

To appear in:

Journal of Physics and Chemistry of Solids

Received Date: 12 February 2018 Revised Date:

29 May 2018

Accepted Date: 25 June 2018

Please cite this article as: E. Jain, G. Pagare, S. Dubey, R. Sharma, Y. Sharma, High pressure phonon and thermodynamic properties of Ru based intermetallics: A DFT investigation, Journal of Physics and Chemistry of Solids (2018), doi: 10.1016/j.jpcs.2018.06.036. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Electron Localization Function (ELF) within 110 plane for (a) RuSc (b) RuTi

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High Pressure Phonon and Thermodynamic Properties of Ru Based Intermetallics: A DFT Investigation Ekta Jain1*, Gitanjali Pagare1*, Shubha Dubey1, Ramesh Sharma2 and Yamini Sharma2 1

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Department of Physics, Sarojini Naidu Government Girls P. G. Autonomous College, Bhopal462016, India 2 Theoretical Condensed Matter Physics Laboratory, Dept. of Physics, Feroze Gandhi College, Raebareli-229001, India

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ABSTRACT

First-principles calculations have been carried out to investigate the structural, electronic,

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phonon and thermodynamic properties of RuX (X = Sc and Ti) intermetallic compounds within the framework of density functional theory. The ground state properties such as lattice parameter (a0), bulk modulus (B) and pressure derivative of bulk modulus (B') have been determined and found in accordance to the available experimental and theoretical results. The electronic

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investigation presents metallic character for these compounds. The electron localization function confirms the dominant ionic bonding between Ru and X. The ductility of these compounds is predicted due to zero frequency gap between optic and acoustic branches of phonon dispersion

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curves. The phonon dispersions and phonon density of states indicate the stability of these compounds in B2-type (CsCl) crystal structure at 0 GPa and 20 GPa. Further we have for the first

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time, predicted the variation of various thermodynamic parameters including Debye temperature, Grüneisen parameter and thermal expansion for these compounds in the temperature range of (02000 K) at different values of pressure (0-100 GPa).

Keywords: Electron localization function, Bader charges, Optical and acoustic modes, Phonon density of states, Thermodynamical properties, High pressure and temperature

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*

Corresponding author: Department of Physics, Sarojini Naidu Government Girls P. G. Autonomous College, Bhopal-462016, India E-mail address: [email protected](Dr. Gitanjali Pagare) Contact No. - +91 9826335268 * 2. Corresponding author: Department of Physics, Sarojini Naidu Government Girls P. G. Autonomous College, Bhopal-462016, India E-mail address: [email protected](Dr. Ekta Jain) Contact No. - +91 9752795241

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1.

1. Introduction

The ruthenium based intermetallics has been a subject of immense research interest due to

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their attractive high temperature physical properties such as high melting temperatures, lower specific gravity, larger temperature strength, good oxidation resistance, greater ductility and high

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thermal conductivity [1-7]. Ruthenium is the most reactive element in platinum group-metals, which shows an excellent catalytic activity [8]. It is extensively used as an alloying agent in the chemical and electronics industries [9]. Fleisher et. al. [7] have performed an extensive experimental search to identify such binary composition which may be strong at progressively

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high temperature so that they might be used in engines to increase their efficiency and performance. They identify the four compounds i.e. AlRu, IrNb, RuSc and RuTa, and each compound has a platinum group metal (PGM) as one of its constituent atoms. In spite of the high

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cost, the favorable properties of PGM alloy motivated us for the further investigations for RuSc and RuTi intermetallic compounds. This study will serve as a reference for other PGM (Rh, Os,

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Ir and Pd) and their intermetallics may display similar properties lower cost and may fulfill the commercial requirements.

The structural, electronic, elastic, phonon and thermodynamic properties of Ru intermetallics have been studied by several groups, employing different theoretical and experimental schemes [10-18]. Jain et. al. [10] have comprehensively focused on phase stability, electronic structure, elastic, mechanical and thermal properties of RuX (X = Sc, Ti and V and Zr) intermetallic

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compounds using full potential linearized augmented plane wave (FP-LAPW) method within DFT. Mehl et. al. [11] have reported first-principles calculations within the framework of local density approximation (LDA) for elastic properties of many binary compounds including RuAl

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and RuZr using density functional theory. They concluded that these compounds exhibit high melting temperatures and large elastic constants and are found to be ductile in nature. Nguyen et. al. [12] computed the electronic structure, phase stability and elastic moduli of AB-type

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transition metal aluminides including RuAl in twelve different structures using FP-LMTO method with local-density approximation (LDA). Ab-initio FP-LAPW calculation has been

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performed by Novakovic et. al. [13] to investigate structural stability of some CsCl structured HfTM (TM = Co, Rh, Ru, Fe) compounds.

The complete phonon dispersion curves are necessary for a microscopic understanding of the lattice dynamics. The structural, thermo-elastic and lattice dynamical properties of the platinum

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metal Rh-base alloys have been reported by Surucu et. al. [14] by using the Vienna ab-initio simulation package (VASP) based on the density functional theory. In Ref. [15-18], the lattice dynamical properties for many binary compounds like AlSc, MgSc, ZrRu, ZrZn, Sc-TM (TM =

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Ag, Cu, Pd, Rh, Ru) and IrX (X = Al, Sc, and Ga) have been investigated in B2 phase. From the above literature it is observed that, there is still need of structural, electronic,

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phonon and thermodynamical properties of RuX (X = Sc and Ti) compounds. Phonon properties of solids are significant as they are closely associated with various fundamental solid-state properties, such as thermal expansion, specific heat, electron-phonon interaction and thermal conduction of the lattice. In this regard, we have paid attention on ambient and high pressure study of phonon dispersion and corresponding phonon density of states to provide additional information to the existing data in the literature. We have also presented for the first time a wide

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variety of structural parameter like volume, bulk modulus, heat capacity at constant volume and pressure, entropy, Debye temperature, Gruneisen parameter, thermal expansion and also examined their behavior on different temperature (0-2000 K) and pressure (0-100 GPa) range. In

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addition, we first time analyze the electron localization function and Bader charges to identify the nature of bonding of these intermetallics. The paper is organized in the following manner: the second part is covered with the computational details; the third part of paper contains the detailed

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analysis of results and discussion. Finally a brief conclusion is given in the last section.

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2. Method of Calculation

We have calculated the phonon spectra and thermodynamical properties of RuX (X = Sc and Ti) intermetallic compounds using first-principles calculations based on full potential linearized augmented plane wave (FP-LAPW) method within the framework of DFT. For lattice dynamical properties, we have used PHONON software [19]. PHONON allows one to perform detailed

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analysis of the transverse and longitudinal optic modes within the approach of the supercell by using an external ab-initio program. In above calculation we have created an interface with

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WIEN2k code [20]. The generalized gradient approximations (GGA) with the exchange correlation function of Perdew et. al. [21] is used. The convergence is achieved by expanding the

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basis function up to RMT*Kmax = 7.0, where RMT is the smallest atomic radius in the unit cell and Kmax gives the magnitude of the largest k vector in the plane wave expansion. The spherical harmonics inside the muffin-tin are taken with an angular momentum lmax=10 while the charge density is Fourier expanded up to Gmax=12 (a.u.)-1. The self-consistent calculations are converged when the total energy of the system is stable within 10-4 Ry. Energy to separate core and valence states is -7.0 Ry. A dense mesh of 10x10x10 k points in tetrahedral method [22] has been employed for the Brillouin zone integration. We have computed the phonon dispersion curve and 4

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phonon density of states for these compounds from Hellmann-Feymann (HF) forces, resulting from a displacement of certain atoms simultaneously from the equilibrium position within supercell of sufficient size. The finite force displacement is fixed at 0.01 Å. For thermodynamic

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calculations, the quasi-harmonic Debye model [23] has been implemented. Further, for electron localization function (ELF) and the Bader charge analysis we have used Bader Aim Concept, as

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implemented in the VASP Code [24].

3. Result and Discussion

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3.1 Structural Properties

As a starting point to understand the physical properties, we have performed the structural optimization for RuX (X = Sc and Ti) in B2-type (CsCl) crystal structure under the minimum condition of the total energy using equation of states (EOS) by Birch-Murnaghan [25]. We have obtained the lattice constant (a), bulk modulus (B) and first-order pressure derivative of the bulk

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modulus (B'), tabulated and compared in Table 1. On inspection of this Table, we found a satisfactory agreement of lattice constant between our calculated value and available experimental [26] and theoretical results [27]. The enthalpies of formation (∆EF) for these binary

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compounds have also been calculated and presented in Table 1. The ground state values are

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found to be -79.45 and -156.80 KJ/mol for RuSc and RuTi respectively. The negative values of the formation energy imply the exothermic nature of the reaction and hence all the intermetallic compounds are found to be thermodynamically stable in B2-type (CsCl) crystal structure. The ∆EF for RuTi is found to be the highest, which shows that this is the more stable compound as compared to RuSc.

3.2 Electronic Properties 3.2.1 Electronic Band Structure and Density of States 5

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In order to get insight into bonding mechanism of RuX (X = Sc and Ti) intermetallic compounds, the non spin polarized self consistent electronic band structures along the principal symmetry direction in B2-phase are calculated and plotted in Fig. 1(a)-1(b). The Fermi level is

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set at 0 eV. One can see from the Fig. 1 that the bands are crossing from valence band to conduction band for both the compounds, indicating that there is no gap between these two regions, and that these compounds are metallic in nature. The lowest lying bands are due to ‘s’

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like state of X (X = Sc and Ti) at Γ-point -5.8 eV and -7.4 eV for RuSc and RuTi respectively. The bands observed just above the ‘s’ like states, originating from -2.0 eV and -3.6 eV and the

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bands observed just below the Fermi level from 0 eV and -1.4 eV are mainly due to ‘d’ like state of Sc and Ti atom for RuSc and RuTi respectively. These bands also cross the Fermi level and mainly responsible for the metallicity of these compounds.

To have an explicit understanding of interaction among different orbitals of atoms, the total

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and projected density of state (DOS) for RuX (X = Sc and Ti) are obtained using PBE-GGA and shown in Fig. 2(a)-2(b) at ambient condition. The overall DOS profile are quite similar for both the RuX compounds except their contributions at the Fermi level. The valence region is mainly

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occupied with ‘d’ like state of Ru atom. The peaks of this state are observed in the energy range between -4 eV to 0 eV and -5 eV to -1 eV for RuSc and RuTi respectively. In the conduction

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band the peaks are mainly due to ‘d’ like state of X (X = Sc and Ti) atom in the energy range 2 eV to 5 eV and 0 eV to 3 eV for RuSc, and RuTi respectively. The metallic behaviour in RuSc is noticed mainly due to ‘d’ like states of Ru while for RuTi the metallicity is observed due to hybridized ‘d’ like states of Ru with ‘d’ like states of Ti at the Fermi level. The DOS values at Fermi level are found to be finite i.e. 1.94 and 0.42 states/eV/F.U. (see Table 1) for RuSc and

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RuTi respectively which verifies the metallic nature of these compounds. The DOS value of RuSc which indicates higher metallicity of this compound at the Fermi level. For both the RuX intermetallic compounds the electronic specific heat coefficient (γ) which is

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the function of density of states is calculated and summarized in Table 1. The electronic specific heat coefficient is proportional to the total DOS at the Fermi level N(EF) and is given by

π N (E )K 2

F

3

2 B

, where N(EF) is the DOS at Fermi level and KB is the Boltzmann’s constant.

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γ=

The obtained values of N(EF) enable us to calculate the bare electronic specific heat coefficient.

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We could not compare the values of γ due to the lack of any reference data.

3.2.2 Electron Localization Function and Bader Charge Analysis

The concept of electron localization function (ELF) was first given by Becke and Edgecombe [28]. The ELF analysis is one of the ways to measure the possibility of distribution of paired

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electrons in solids compared to plain charge distributions. Basically it the measure of the probability of finding an electron in the neighborhood of a reference electron with the same spins and situated at a given point. ELF measures the extend up to which the spatial localization of

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reference electron is done. It provides a mapping method of electron pair probability in multielectronic systems. The ELF in (110) plane for RuX compounds is depicted in Fig. 3(a)-

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3(b). From Fig. 3, it is seen that isosurface with highest value of ELF (in red color) appears around the location of X (X = Sc and Ti) whereas isosurface has very low value (in blue color) around Ru atom, which confirms the dominant ionic bonding in RuX intermetallics. Between the Ru and X atoms, the ELF does not show any maximum value, and this indicates that there is no dominant covalent-type bonding interaction present between these atoms. However, nonspherical

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distribution of ELF at the X site and small but finite value of ELF in between Ru and X reflect the presence of non-negligible hybridization interaction between these atoms. We performed a Bader charge analysis of these systems for quantitative characterization of

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chemical bonding in solid. In the Bader-charge analysis, each atom of a compound is surrounded by a surface called Bader region that runs through the minima of the charge density and the total charge of the atom is determined by integration within the Bader region. We observe definitive

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charge transfer from X (X = Sc and Ti) i.e. -1.3431e and -1.2857e respectively to Ru atom. The gain of -ve charge is consistent with higher electronegativity of Ru atom. This is confirmed by

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the ELF plot, where it is clear that the Ru is activated with a net negative charge localized on the X (X = Sc and Ti) atoms. Table 2, presents the charges at each atomic site for Ru calculated according to the Bader charges. As expected, Sc and Ti atoms donate electrons while Ru atoms accept them. Based on the amount of the transferred charges, one can conclude that chemical

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bonding is of dominant ionic type.

3.3 Phonon Dispersion Curve

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The Phonon dispersion curves (PDC) are calculated at 0 GPa and 20 GPa and presented in Fig. 4(a)-4(d) for RuX (X = Sc and Ti) compounds along the path that contains the highest

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symmetry points in the Brillouin zone (BZ), namely Γ→X→M→Γ→R→M. The dynamical properties of these intermetallics are calculated in B2-type (CsCl) crystal structure with space group symmetry Pm3m (No. 221) using FP-LAPW method to have insight on changes that occur in phonon dispersion curves and phonon associated thermal properties. These materials contain two atoms per primitive cubic unit cell, in which Ru is positioned at (0, 0, 0) and X is positioned at (0.5, 0.5, 0.5). Due to the symmetry the distinct number of branches is reduced along the high symmetry directions Γ→X and Γ→R→M. As one can notice from the Fig. 4 that at 0 GPa for 8

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both the RuX compounds there is no frequency gap between optical and acoustic phonon modes or they intersect each other at more than one BZ point, indicating that these compounds are ductile in nature. This interpretation can be compared with the phonon dispersion curves for

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NiAl, YCu and YAg are presented in B2 structure by the Wang et. al. [29], they concluded that in ductile intermetallics YAg and YCu, there is no frequency gap observed between optical and acoustical phonon modes whereas in NiAl this gap is started at 6 THz. There are many papers in

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which the ductility has been analysed by Pugh’s criteria which can be correlated with the phonon dispersion curves such as Arikan et. al. [18, 30] have investigated the phonon dispersion curve

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for HfX (X = Rh, Ru and Tc) and IrX (X = Al, Sc and Ga) in B2 phase and predicted the ductility of these intermetallics using Pugh’s criteria (B/GH). On observing the phonon dispersion curves of these compounds, it confirms that there is very less or no frequency gap between both the phonon branches for all these materials. At high pressure the frequency gap between both the

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modes is slightly increased only at M point. The frequency gap can be directly associated with the mass ratio of the constituent atoms. Higher mass ratio indicates larger frequency gap. In case of our compounds this ratio is found to be lowest. The frequency of phonon modes for both the

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RuX compounds is positive throughout the BZ, which clearly shows the dynamical stability of these compounds in B2-type (CsCl) crystal structure at ambient and high pressure. The

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vibrational modes at different Brilliouin zone points for RuX at ambient and high pressure and at 300K temperature is presented in Table 3. For RuX, the optical phonon modes are degenerated at zone centre Γ point and again split in LO (longitudinal optical) and TO (transverse optical) branches from zone centre, this splitting indicates the long range coulomb interaction [31]. The LO and TO modes degenerate with frequency nearly 41 and 42 THz at zero GPa for RuSc and RuTi respectively while at high

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pressure these phonon frequency shifted upward at around 52, 50 THz. This might be due the smaller lattice parameter or bond length at high pressure because shorter bond length leads to larger force constant, resulting in higher vibrational frequencies [32]. The highest phonon

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frequency for optical modes is calculated to be 54.79 THz for RuSc and 67.81 THz for RuTi at X point for 0 GPa.

The acoustic phonon modes are smooth and almost similar for both the RuX compounds. For

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acoustic branch the softening of LA (longitudinal acoustic) and TA (transverse acoustic) modes are observed from Γ-M point for these compounds. A clear splitting of TA1 and TA2 branches is

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seen for both the RuX intermetallics at X point, which again degenerate at zone centre Γ point at 0 pressure. A minor softening in LA modes is also seen at high pressure for these compounds. One can see from these Figs. that LA and TO phonon modes cross each other at more than one point in BZ mainly due to their mass ratio, which is significantly small. We have compared our

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results of the phonon dispersion curves and phonon density of states for RuSc with available theoretical result of Arikan et. al. [16] and found a good agreement among the studies. But the results for RuTi could not be compared due to lack of experimental and other theoretical data.

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The phonon density of states (PhDOS) associated with phonon dispersion curve is presented at 0 GPa and 20 GPa in Fig. 5(a)-5(d) for both the RuX compounds. For RuSc, three main peaks

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are observed in PhDOS. These peaks are situated between 20-25, 25-30, 45-50 THz, which is due to TA, LA, LO phonon modes respectively at ambient pressure. The frequency ranges for the peaks in RuTi are 30-35, 35-40, 50-55 THz which is due to TA, LA, LO vibrational modes. At high pressure these peaks are slightly shifted to higher frequency range for both studied compounds. One can see from Fig 4 that at higher frequency the density of states are composed of X (X = Sc and Ti) states, because its atomic mass is comparably lighter than Ru atom.

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3.4 Thermodynamical Properties The thermodynamic properties of a solid are directly related to its phonon structure. The

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calculated phonon density of states (PhDOS) is used to determine the heat capacity, entropy, free energy etc. of a crystal. The study of thermodynamic properties of materials under high pressure and high temperature environment is essential in order to extend our knowledge about their

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specific behavior. The thermodynamic properties like volume, bulk modulus, heat capacity at constant volume and pressure, entropy, Debye temperature, Gruneisen parameter and thermal

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expansion are determined in the temperature range 0-2000 K, where the quasi-harmonic Debye model remains fully valid [23]. The pressure effect is also taken into account in the 0-100 GPa range. We have also presented these thermodynamic properties in Table 4 at ambient pressure and room temperature. The motive of above calculation is to have a comprehensive insight into

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the influence of phonon exerted on these properties. These quantities have been calculated using the following equations. In this model the Gibbs function takes the form: ∗

, ,

=

+

+

,

(1)

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where E(V) is the total energy per unit cell, P(V) corresponds to the constant hydrostatic

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pressure, θ(V) is the Debye temperature and Fvib is the vibration term written as; [

;

=

+ 3

$

1 − # % & − ' ( )*

(2)

where D (θ/T) represents the Debye integral, N is the number of atoms per formula unit and KB is Boltzmann’s constant. For an isotropic solid, the Debye temperature θD is expressed as [33, 34] +

=

,

-.

/61

2

3 4

3 6

5 7 8 9

:

;

(3)

In the above equation BS is the adiabatic bulk modulus and M is the molecular mass per unit cell, the bulk modulus is expressed by 11

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<= =

(

>4 ? @ >@ 4

)

(4)

The non-equilibrium Gibbs function G*(V; PT) can be minimized with respect to volume V; )

>@

=0

B,

(5)

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>A∗ @;B,

(

The solution to (equ. 5) helps one to get thermodynamic quantities like thermal expansion α heat capacity at constant volume CV and heat capacity at constant pressure CP is respectively given by

D=

EFG

%@

I4' K

L

−

M L/

QR

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H@ = 3

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[33],

$

L P3 O%

HB = H@ 1 + SD

(6)

(7) (8)

In (equ. 8) ς represents the Gruneisen parameter, which is approximated as

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ς= −

>TU L @ >TU@

(9)

The entropy S is described as:

4' ( L ) − 3

$L

1 − # % &*

(10)

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V=

The variation of unit cell volume with temperature range (0-2000 K) at different pressure (0-

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100 GPa) is presented in Fig. 6(a)-6(b) for RuX (Sc and Ti) compounds. It is seen that volume slightly increases in accordance with temperature. But when higher pressure is applied, it decreases, can be accredited to the fact that the interlayer in atom comes closer and the interaction between them becomes stronger. Fig. 7(a)-7(b) shows the temperature dependence of bulk modulus at different pressure 0, 20, 40 60, 80, 100 GPa. The effect of temperature on isothermal bulk modulus BS is expected as very small. While for both compounds the values of bulk modulus is obtained larger at highest pressure. 12

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The Heat capacity does not provide a fundamental knowledge of vibration properties but also is an essential parameter for many applications in condensed matter and material physics. Temperature dependence of the heat capacity Cv provides the complete picture of vibrations of

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atoms and has been able to be determined only experimentally for a long time past [35]. In Fig. 8(a)-8(b) and Fig. 9(a)-9(b), we demonstrated the variation of heat capacities at constant volume (Cv) and pressure (Cp) at different ranges of temperature and pressure defined above for both the

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RuX intermetallics. At low temperature < 500K, the shapes of curves for Cv and Cp are similar but at higher temperatures Cv reaches close to Dulong-Petit limit [36]. The saturation value of Cv

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at higher temperature is might be due to superposition of harmonic effect. While Cp deviates from the trend of Cv, and it becomes linear at higher temperature range. The Cv at ambient pressure and room temperature for RuSc and RuTi is 45 and 43 J/molK, respectively. Fig. 10(a)10(b) depicted the variation of entropy, S with temperature at different pressure value. One may

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notice that entropy increases with increasing temperature, because as the energy is imposed into a system, excites the molecules and the amount of random activity. Fig. 10 also provides a clear understanding that it decreases as pressure increases. We have also presented the value of

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entropy at ambient pressure and room temperature in Table 4. In quasi-harmonic Debye model, the Debye temperature (θD) and Gruneisen parameter, ς are

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two key quantities, which are very sensitive to the vibrational modes. The temperature dependence of Debye temperature for both the RuX compounds follow similar trend. In Fig. 11(a)-11(b), one can see that as the temperature increases, Debye temperature decreases with a moderate pace, and this decrease found a linear fashion. The variation of our calculated Gruneisen parameter (ς) with temperature in Fig. 12(a)-12(b) is in such a way that at lower temperature its value remains almost constant and then slightly increases with increasing

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temperature. On the other hand one can notice a decrease in the values of ς with pressure. Further, In Fig. 13(a)-13(b), we have plotted the variation of thermal expansion coefficient, α with temperature and pressure. Preliminary α increases with fast pace at low temperature and at

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higher temperature it increasing rate gradually decreases. On the other side it decreases with the increasing value of pressure, and it takes the highest values at all temperature ranges at lowest (P = 0 GPa) pressure.

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Some of the above mention properties like volume, bulk modulus, Debye temperature and Gruneisen parameter are calculated from quasi harmonic approximation (QHA) and FP-LAPW

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method and compared in Table 4. One can see from this Table that both the theoretical approaches are in good agreement which clearly indicates the reliability of this work. Unfortunately, there is no experimental or other theoretical data on these thermodynamical

3.5 Melting Temperature

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properties of studied compounds for comparison with the present results.

The melting temperature is an important physical property of a material that can be a

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prediction of its identification and also used as an indicator of its purity. It can be correlated with the force of attraction between the molecules of a compound. Stronger intermolecular

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interactions results higher melting temperatures. Fine et. al. and Dar et. al. [37, 38] have obtained an empirical relationship between melting temperature and elastic constant. They have shown that melting temperature may be used for estimation of C11 in cubic system. It may be noted here that in obtaining those correlations, room temperature elastic constants were used. It can be calculated by employing the following empirical formula: W

= [553Y + 5.91Y/

14

\ H]] ± 300Y

(11)

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In Table 5, we have presented and compared our calculated and experimental [39] melting temperature for these systems and found a reasonable agreement between these values. The significant higher melting temperature of these compounds is expected to lead to the realization

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of a useful high-temperature structural material. In this table, we have also incorporated the second order elastic constants and sound wave velocities for these compounds.

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4. Conclusions

In summary full-potential linearized augmented plane wave method based upon density

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functional theory within generalized gradient approximation has been used to obtain the structural, electronic and phonon properties while quasi harmonic approximation has been implemented to investigate these RuX (X = Sc and Ti) intermetallic compounds for their thermodynamic character. The calculated ground state properties such as lattice constants, bulk modulus and its pressure derivative for both the compounds are found in a good agreement with

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the available experimental and theoretical results. The negative formation energy implies thermodynamic stability for these compounds in B2-type (CsCl) crystal structure. The computed

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band structure and density of states show metallic nature for both the compounds. The electron localization function presents bonding nature as ionic for both the compounds. The calculated

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phonon dispersion curve and corresponding phonon density of states also approve the dynamical stability of these intermetallic compounds at ambient and high pressure in B2 phase due to the absence of any imaginary phonon frequency. The optical and acoustic modes in dispersion curve are not separated by a frequency gap, which emphasize the ductile character of these compounds. The density of states at higher frequency are mainly composed of ‘X’ states, as expected because its atomic mass is comparably lighter than Ru atom. The temperature dependence of volume, bulk modulus, heat capacity at constant volume and pressure, entropy, Debye temperature, 15

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Gruneisen parameter and thermal expansion in temperature range (0-2000 K) at different pressure (0-100 GPa) has been presented and analyzed. A comparative study of some of these properties like volume, bulk modulus, Debye temperature and Gruneisen parameter has also been

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performed by two different theoretical approaches i.e. QHA and FP-LAPW and found a good agreement between the two. The contribution of present paper is mainly concerned about electron localization function, Bader charge analysis, phonon dispersion curve, phonon density

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of states and phonon exerted thermodynamic properties at higher range of temperature and pressure for high temperature structural use of these materials, which have not been considered

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so far.

References

P. Lazar, R. Podloucky, Phys. Rev. B 73 (2006) 104114.

[2]

P.Gumbsch, R. Schroll, Intermetallics 7 (1999) 447.

[3]

S. K. Pasha, M. Sundareswari, M. Rajagopalan, Physica B 348 (2004) 206-212.

[4]

E. Jain, G. Pagare, S. S. Chouhan, S. P. Sanyal, Comp. Material Sci. 83 (2014) 64-69.

[5]

H. fu, X. Li, W. Liu, Y. Ma, T. Gao, X. Hong, Intermetallics 19 (2011) 1959-1967.

[6]

K. A. Gschneidner Jr., M. Ji, C.Z. Wang, K.M. Ho, A.M. Russell, Y. Mudryk, A.T.

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[1]

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Becker, J. L. Larson, Acta Mater. 57 (2009) 5876-5881. [7]

R. L. Fleischer, Platinum Metal Rev. 36 (1992) 138-145.

[8]

F.R. Hartley (Ed.), The Occurrence, Extraction, Properties and Uses of the Platinum

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Group Metals, in the Chemistry of the Platinum Group Metals, Recent Developments, Elsevier, Amsterdam, 1991 pp 9-31.

[9]

P. J. Loferski, Minerals Yearbook-Platinum Group Metals (US Geological Survey, Washington, DC, 2008).

[10] [11]

E. Jain, G. Pagare, S. S. Chouhan and S. P. Sanyal, Intermetallics 54, (2014) 79-85.

M. J. Mehl, B. M. Klein, D. A. Papaconstatopoulos, Intermetallic compounds: Principles and Practice New York: John Wiley and Sons Ltd vol. 1 (1995) 195-210.

[12]

D. Nguyen-Manh and D. G. Pettifor, Intermetallics 7(1999) 1095-1106.

16

ACCEPTED MANUSCRIPT

[13]

N. Novakovic, N. Ivanovic, V. Koteski, I. Radisavljvic, J. Belosevic-Cavor, B. Cekic, Intermetallics 14 (2006) 1403-1410. G. Surucu, K. Colakoglu, E. Deligoz, H. Ozisik, Intermetallics 18 (2010) 286-291.

[15]

S. Ugur, N. Arikan, F. Soyalp, G. Ugur, Comp. Material Sci. 48 (2010) 866-870.

[16]

N. Arikan, S. Ugur, Comp. Material Sci.47 (2010) 668-671.

[17]

N. Arikan, U. Bayhan, Physica B 406 (2011) 3234-3237.

[18]

N. Arikan, Z. Chairifi, H. Baaziz, S. Ugur, H. Unver, G. Ugur, J. of Phys. and Chem. of Solids 77 (2015) 126-132.

RI PT

[14]

K. Parlinski, Z. Q. Li and Y. Kawazoe, Phys. Rev. Lett.78, 4063 (1997).

[20]

P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka and J. Luitz, Techn. Universit¨at

M AN U

Wien, Austria, ISBN 39501031-1-2, 2001.

SC

[19]

[21]

J. P. Perdew, K. Burke and M. Ernzerhop, Phys. Rev. Lett. 77 3865-3868 (1996).

[22]

H. J. Monkhorst, J. D. Pack, Phys. Rev. B 13 (1976) 5188-5192.

[23]

M. A. Blanco, E. Francisco, V. Luania, Gibbs isothermal-isobaric thermodynamics of solids from energy curves using a quasiharmonic Debye model, Comput. Phys. Commun. 158 (2004) 57-72.

G. Kresse, J. Hafner, ab initio molecular dynamics for liquid metals, Phys. Rev. B 48

TE D

[24]

(1993) 13115, Phys. Rev. B 49 (1994) 14251; G. Kresse, J. Furthmüller, “Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set”, Comput. Mat. Sci. 6 (1996) 15; Efficient iterative schemes for ab initio total-

EP

energy calculations using a plane-wave basis set”, Phys. Rev. B 54 (1996) 11169. F. Birch, J. Appl. Phys. 9 (1938) 279-288.

[26]

F. A. Shunk, Constitution of binary alloys, McGraw-hill Book company (New-York) v.

AC C

[25]

1-2 (1971).

[27] [28] [29] [30]

M. A. Baranov, EPhTJ 1 (2006) 49-62. A. D. Becke and K. E. Edgecombe, J. Chem. Phys. 92 (1990) 5397. R. Wang, S. Wang, X. Wu, A. Liu, Intermetallics 19 (2011) 1599-1604.

A. Iyigör, M. Özduran, M. Ünsal, O. Örnek and N. Arikan, Philosophical Magazine Letters, DOI: 10.1080/09500839.2017.1290292, (2017) 1-8.

[31]

B. Roondhe, D. Upadhyay, N. Som, S. B. Pillai, S. Shinde and P. K. Jha, J. of. Elec. Mater. DOI:10.1007/s11664-5247-1 (2016) 17

ACCEPTED MANUSCRIPT

[32]

H. Peng, C.L. Wang, J.C. Li, R.Z. Zhang, M.X. Wang, H.C. Wang, Y. Sun and M. Sheng, Phys. Lett. A 374,3797-3800 (2010).

[33]

A. Otero-de-la-Roza, D. Abbasi-Perez, V Luaea, Gibbs2: a new version of quasiharmonic

Comput. Phys. Commun. 182 (2011) 2232-2248. [34]

RI PT

model code, II. Models for solidstate thermodynamic features and implementation,

M. A. Blanco, P. A. Martin, E. Frncisco, J. M. Recio, R. Franco, Thermodynamical properties of solids from microscopic theory: applications to MgF2 and Al2O3, J.Mol.Struct.Theochem.368 (1996) 245-255. M.J. Mehl, Phys. Rev. B 47 (1993) 2493.

[36]

A. T. Petit, P. L. Dulong, Study on the measurement of specific heat of solids, Ann.

M AN U

Chim. Phys. 10 (1819) 395-413.

SC

[35]

[37]

M. E. Fine, L. D. Brown, H. L. Marcus, Scr. Metall. 18 (1984) 951-956.

[38]

S. A. Dar, V. Srivastava, U. K. Sakalle, J. Electronic Materials 46 (2017) 6870-6877.

[39]

M. Hansen, Constitution of Binary Alloys, McGraw Hill, New York (1958); R. P. Elliot, Constitution ofBinary Alloys-First Supplement, McGraw Hill, NewYork (1965); F. A. Shunk, Constitution of Binary Alloys-Second Supplement, McGraw Hill, Gew

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York(1965); W. G. Moffatt, Binary Phase Diagrams Hand-book, General Electric,

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Schenectady, New York (1976).

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ACCEPTED MANUSCRIPT Table 1 The calculated ground state and electronic properties for RuX (X = Sc and Ti) at ambient condition using PBE-GGA.

a

Ref [26] Ref [27]

b

RuTi 3.0865 3.06a 3.102b 220.97 4.07 -156.80 0.42 1.01 3.0865 2.630

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RuSc 3.2138 3.203a 3.288b 143.65 4.22 -79.45 1.94 4.59 3.2138 2.7832

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Calculated Parameters ao (Å) Present Experimental Other Theory B (GPa) B' ∆EF (KJ/mol) N(EF) (States/eV) γ (mJ/mol K2) Bond Length (Å) Ru-Ru Ru-X

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Table 2. Calculated Bader charges, valence charge and charge transfer inside regions.

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RuSc RuTi Calculated Ru Sc Ru Ti Parameters Valence 15.3431 9.6569 15.2857 10.7143 Charges Charge -1.3431 1.3431 -1.2857 1.2857 Transfer

Table 3 Vibrational modes at different Brilliouin zone points for RuX (X = Sc and Ti) at ambient and high pressure and at 300K temperature.

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Brilliouin Zone Points Γ

Calculated Modes (THz)

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LTO [T1u (I)] TA LA TO1 TO2 LO TA1 TA2 LA TO1 TO2 LO LTA LTO

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Frequency (THz) RuSc RuTi 0 GPa 20 GPa 0 GPa 20 GPa 41.302 51.578 42.383 50.452 23.23 28.193 31.304 34.838 28.057 36.618 32.314 37.313 29.919 37.824 41.163 45.378 28.057 36.618 32.314 37.313 54.798 67.447 69.382 76.502 35.568 39.731 18.726 20.443 31.686 35.004 29.04 33.783 36.974 39.731 48.286 60.266 53.921 62.758 29.331 36.042 36.974 42.363 48.286 60.266 53.921 62.758 24.323 32.818 41.721 46.369 46.624 56.441 50.478 56.23

ACCEPTED MANUSCRIPT Table 4 A comparative study of the calculated volume (V), bulk modulus (B), heat capacity at constant volume (Cv), heat capacity at constant pressure (Cp), entropy (S), Debye temperature (ƟD), Gruneisen parameter (ς) and thermal expansion (α) for RuX (X = Sc and Ti) compounds using QHA and FP-LAPW method.

V (Bohr3) B (GPa) Cv (J/molK) Cp (J/molK) S (J/molK) ƟD (K)

RuTi 0K 300K 199.38 200.07 198.43 216.96 212.86 220.97 0 43 0 43.5 0 42.1 529.05 525.68 501.99 1.82 1.83 1.62 0 2.07

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RuSc 0K 300K 225.01 226.56 223.99 142.99 136.52 143.65 0 45 0 46.2 0 50.4 441.36 434.37 426.90 2.31 2.34 1.58 0 3.81

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Approximations Used QHA FP-LAPW QHA FP-LAPW QHA QHA QHA QHA FP-LAPW QHA FP-LAPW QHA

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Calculated Parameters

α (10-5K-1)

Table 5 Calculated second order elastic constants (C11, C12, C44), sound wave velocities and melting temperature for RuX (X = Sc and Ti) at ambient condition.

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Calculated Parameters C11 (GPa) C12 (GPa) C44 (GPa) vl (m/s) vt (m/s) vm (m/s) Tm ± 300K Pre. Exp. Pre. - Present, Oth. Exp. - Experimental a Ref [39]

RuSc

RuTi

256 87 75 5836 3290 3660 2066 2473a

387 135 109 6659 3707 4129 2840 2423a

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Fig. 1(a)-1(b) Electronic Band Structure for RuX Compounds at ambient condition

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Fig. 2(a)-2(b) Total and Partial Density of States for RuX Compounds at ambient condition

Fig. 3(a)-3(b) Electron Localization Function (ELF) within 110 plane for (a) RuSc (b) RuTi

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Fig. 4(a)-4(d) Phonon Dispersion Curves for RuX Compounds at 0 GPa and 20 GPa

Fig. 5(a)-5(d) Phonon Density of States for RuX Compounds at 0 GPa and 20 GPa

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Fig. 6(a)-6(b) Variation of Volume with Temperature for RuX Compounds

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Fig. 7(a)-7(b) Variation of Bulk Modulus with Temperature for RuX Compounds

Fig. 8(a)-8(b) Variation of Heat Capacity, Cv with Temperature for RuX Compounds

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Fig. 9(a)-9(b) Variation of Heat Capacity, Cp with Temperature for RuX Compounds

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Fig. 10(a)-10(b) Variation of Entropy with Temperature for RuX Compounds

Fig. 11(a)-11(b) Variation of Debye Temperature with Temperature for RuX Compounds

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Fig. 12(a)-12(b) Variation of Gruneisen Parameter with Temperature for RuX Compounds

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Fig. 13(a)-13(b) Variation of Thermal Expansion with Temperature for RuX Compounds

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Research Highlights

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The negative values of the formation energy and the positive value of frequency in PDC and PhDOS confirms the stability for RuX compounds in B2-type (CsCl) crystal structure.

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The electron localization function confirms the dominant ionic bonding between Ru and X. The density of states at higher frequency are mainly composed of ‘X’ states, as expected because its atomic mass is comparably lighter than Ru atom.

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The calculated and experimental melting temperature shows good agreement.